Volume of a Right Circular Cone

Date: 10/07/2001 at 19:45:43
From: Jeffrey A Dozier
Subject: Volume of a right circular cone

Using calculus, derive the formula for the volume of a right circular 
cone with a radius of r and height h. I seem to remember that there 
is a way using calculus to create a proof showing that the volume of 
a right circular cone is: pi r^2 h/3 .

Date: 10/07/2001 at 23:11:54
From: Doctor Paul
Subject: Re: Volume of a right circular cone

Draw an x-y coordinate system. In the first coordinate, draw a line 
from the origin to some point (h,r). This represents the line 
y = r/h * x.

We want to rotate this line about the x axis and compute the volume of 
the solid obtained thereby. Note that the resulting solid would be a 
right circular cone with height h and radius r.

The formula for the volume of a solid of revolution is given by:

Pi * integral of f(x)^2 dx

so what we have is:

Pi * integral from 0 to h of ((r/h)*x)^2) dx

square the integrand and pull out the constant r^2/h^2

then we have:

Pi*r^2/h^2 * integral from 0 to h of x^2 dx

the integral becomes 1/3 * h^3 when evaluted.

When 1/3 * h^3 is multiplied by the constants that were outside the 
integral, we get the desired answer:

1/3 * Pi * r^2 * h

I have been deliberately brief in the hope that you can fill in any 
missing details. If something is unclear, please write back.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/